Structure of a Line Vortex in an Imposed Strain
The velocity of a vortex line depends on its structure, i.e. the shape of the cross-section and the detailed vorticity distribution. As a first step towards an understanding of how the structure depends on the motion and the construction of a valid approximation for the motion of vortex lines in general flow fields, we consider the structure of straight line vortices in a uniform two-dimensional straining field. Two cases are considered in detail, irrotational strain and simple shear. In the first case, it is shown that steady exact solutions of the inviscid equations exist, in which the boundary of the vortex is an ellipse with principal axes at 45° to the principal axes of strain. There are two possible axis ratios provided e/ωo<0.15, where e is the maximum rate of extension and ωo is the vorticity in the core. The stability of the shapes is considered, and it is shown that the more elongated shape is unstable, while the less elongated one is stable to two-dimensional deformations. There are no steady solutions of elliptical form if e/ωo>0.15, and it is believed from some numerical work that in this case the strain field will cause the vortex to break up. For simple shear, there is one steady shape of elliptical form if the shear rotation and vorticity are in the same sense and e′<ωo, where e′ is the rate of shear. The major axis is parallel to the streamlines and the shape is stable to two-dimensional deformations. For shear rotation and vorticity in opposite senses, there are two steady elliptical shapes if e′/ωo<0.21, with major axes perpendicular to the streamlines. The more elongated form is unstable, and the less elongated one is stable. Disturbances of three-dimensional form are also considered in the limit of extremely large axial wavelength.
KeywordsMajor Axis Simple Shear Vortex Line Axis Ratio Point Vortex
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